Answer to Question #203656 in Linear Algebra for rama

Question #203656

Using the the concept of rank of the matrix find whether the following system of equations

are consistent or inconsistent,

i.

4y + z = 0

12x - 5y - 3z = 34

-6x + 4z = 8


1
Expert's answer
2021-06-09T08:44:28-0400

i. Ax=bAx=b is inconsistent (i.e., no solution exists) if and only if rank[A]<rank[Ab].\text{rank}[A]<\text{rank}[A|b].


ii. Ax=bAx=b has an unique solution if and only if rank[A]=rank[Ab]=n.\text{rank}[A]=\text{rank}[A|b]=n.


iii. Ax=bAx=b has infinitely many solutions if and only if rank[A]=rank[Ab]<n.\text{rank}[A]=\text{rank}[A|b]<n.



A=[0411253604]A=\begin{bmatrix} 0 & 4 & 1 \\ 12 & -5 & -3 \\ -6 & 0 & 4 \end{bmatrix}

Swap rows 1and 2


[1253041604]\begin{bmatrix} 12 & -5 & -3 \\ 0 & 4 & 1 \\ -6 & 0 & 4 \end{bmatrix}

R3=R3+(1/2)R1R_3=R_3+(1/2)R_1

[125304105/25/2]\begin{bmatrix} 12 & -5 & -3 \\ 0 & 4 & 1 \\ 0 & -5/2 & 5/2 \end{bmatrix}

R3=R3+(5/8)R2R_3=R_3+(5/8)R_2


The rank of a matrix is the number of nonzero rows in the reduced matrix, so the rank is 3.


Since rank[A]=rank[Ab]=3=n,\text{rank}[A]=\text{rank}[A|b]=3=n, then Ax=bAx=b has an unique solution.

Therefore, the given system of equations is consistent.



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